Shape Signature(2)

 Cumulative angular function. Intuitively, the tangent angles of the shape boundary indicate the change of angular directions of the shape boundary. The change of angular directions is important to human perception.Therefore, shape can be represented by its boundary tangent angles:

where w, a integer, is a jump step used in practice. However,the tangent angle function    q(t) can only assume values in a range of length 2p, usually in the interval of [-pi, pi] or[0, 2pi]. Therefore, q(t) in general contains discontinuities of size 2pi, see Fig. 3(b). Because of this, a cumulative angular function is introduced to overcome this problem. The
cumulative angular function    is the net amount of angular bend between the starting position z(0) and position z(t) on the shape boundary

The acquiring of    is illustrated in Fig. 3(d). As can be seen in Fig. 3(c),   is continuous at places where is multiples of 2pi, although it still has discontinuities at boundary positions where sharp changes occur.
Curvature signature. Curvature is a important boundary feature. It is used widely for shape representation in the literature. Curvature function is given by

Area function. It is known that area changes linearly under affine transformation. Linearity is a desirable property for shape representation, because normalization of linearity is equivalent to scale normalization, which is simple.Therefore, we propose an area signature in attempt to acquire FD invariant to affine distortion.When the boundary points change along the shape boundary, the area of the triangle formed by the two boundary points and the center of gravity also changes (Fig. 5(a)). This forms an area function, which can be exploited as shape representation.For the triangle formed by o, p1 and p2 in Fig. 5(b), its area is given by

For each boundary points, the area of the triangle with 5 degree angle at vertex o is calculated.